Finite Group Databasebeta

← #30744173

Subgroups — #30744173

Order 17152.

432 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560352C211yesmaximal, characteristic, complemented
2#22559955C211yesmaximal, characteristic, complemented
3#22558035C212yesmaximal, complemented
4#22558030C212yesmaximal
5C67×#2861C211yesmaximal, characteristic
6C2×C4×C67⋊C8—21no—
7#16863151—21no—
8#16863151—21no—
9#16863151—21no—
10#16863151C2212yes—
11#16863151C2212yes—
12C2×#125×C67C411yescharacteristic
13C2×#124×C67C411yescharacteristic
14#391×C67C411yescharacteristic
15#391×C67C2211yescharacteristic
16#390×C67C411yescharacteristic
17#390×C67C2211yescharacteristic
18#388×C67—21no—
19#388×C67—21no—
20#388×C67—21no—
21#388×C67—21no—
22C43×C67C2211yescharacteristic
23C22×C67⋊C8—22no—
24C22×C67⋊C8—22no—
25C4×C67⋊C8—24no—
26C22×Q8×C67C2×C411yescharacteristic
27C22×D4×C67C2×C411yescharacteristic
28#125×C67—42no—
29#124×C67—42no—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67C2×C411yescharacteristic
33C2×C4⋊C4×C67C2×C411yescharacteristic
34C2×C22⋊C4×C67—21no—
35C2×C22⋊C4×C67—21no—
36C2×C22⋊C4×C67—21no—
37C2×C22⋊C4×C67—21no—
38C2×C22⋊C4×C67—21no—
39C2×C22⋊C4×C67—21no—
40C2×C42×C67C2×C411yescharacteristic
41C2×C42×C67C2×C411yescharacteristic
42C2×C42×C67D411yescharacteristic
43C2×C42×C67D411yescharacteristic
44C2×C42×C67D411yescharacteristic
45C2×C42×C67D411yescharacteristic
46C2×C42×C67C2311yescharacteristic
47#95×C67—21no—
48#95×C67—21no—
49#95×C67—21no—
50#95×C67—21no—
51C2×C67⋊C8—24no—
52C2×C67⋊C8—42no—
53C2×C67⋊C8—42no—
54C2×C67⋊C8—42no—
55C2×C67⋊C8—42no—
56C2×C67⋊C8—42no—
57C24×C67—21no—
58C2×Q8×C67—22no—
59C2×Q8×C67—41no—
60C2×Q8×C67—41no—
61C2×D4×C67—22no—
62C2×D4×C67—41no—
63C2×D4×C67—41no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22×C411yescharacteristic
72C22×C4×C67C2×D411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C4⋊C4×C67—41no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—42no—
78C22⋊C4×C67—42no—
79C22⋊C4×C67—42no—
80C22⋊C4×C67—42no—
81C22⋊C4×C67—42no—
82C22⋊C4×C67—42no—
83C22⋊C4×C67—42no—
84C42×C67—22no—
85C42×C67—22no—
86C42×C67—22no—
87C42×C67—22no—
88C42×C67—22no—
89C42×C67—41no—
90C42×C67—41no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67C2×C22⋊C411yescharacteristic
99Q8×C67—82no—
100D4×C67—82no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67#10012yes—
127C2×C4×C67#10412yes—
128C2×C4×C67#13112yes—
129C2×C4×C67#13512yes—
130C2×C4×C67#13512yes—
131C67⋊C8—44no—
132C67⋊C8—82no—
133C22×C67—81no—
134C22×C67—81no—
135C22×C67—81no—
136C22×C67—81no—
137C22×C67—81no—
138C22×C67—81no—
139C22×C67—81no—
140C22×C67#40811yescharacteristic
141C22×C67#41011yescharacteristic
142C22×C67C2×#10011yescharacteristic
143C22×C67#41611yescharacteristic
144C22×C67#41611yescharacteristic
145C22×C67C2×#10411yescharacteristic
146C22×C67#42111yescharacteristic
147C4×C67—22no—
148C4×C67—22no—
149C4×C67—22no—
150C4×C67—22no—
151C4×C67—22no—
152C4×C67—41no—
153C4×C67—41no—
154C4×C67—81no—
155C4×C67—81no—
156C4×C67—81no—
157#7656—671nomaximal, Sylow
158C2×C67—161no—
159C2×C67#146911yescharacteristic
160C2×C67#148211yescharacteristic
161C2×C67#149411yescharacteristic
162C2×C67#150011yescharacteristic
163C2×C67#152111yescharacteristic
164C2×C67#153011yescharacteristic
165C2×C67#154311yescharacteristic
166#2861D6711yescharacteristic
167#1839—671no—
168#1744—671no—
169#1272—672no—
170#1270—672no—
171C67#765611yescharacteristic, complemented, Sylow
172C2×#125Dic6711yescharacteristic
173C2×#124Dic6711yescharacteristic
174C2×C4×C8—1341no—
175#391Dic6711yescharacteristic
176#391C2×D6711yescharacteristic
177#390Dic6711yescharacteristic
178#390C2×D6711yescharacteristic
179#388—21no—
180#388—21no—
181#388—21no—
182#388—21no—
183C43C2×D6711yescharacteristic
184#336—672no—
185#336—672no—
186#336—1341no—
187#336—1341no—
188#336—1341no—
189C22×Q8C2×Dic6711yescharacteristic
190C22×D4C2×Dic6711yescharacteristic
191C22×C8—1342no—
192C22×C8—1342no—
193#125—42no—
194#124—42no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic6711yescharacteristic
198C2×C4⋊C4C2×Dic6711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42C2×Dic6711yescharacteristic
206C2×C42C2×Dic6711yescharacteristic
207C2×C42C22⋊D6711yescharacteristic
208C2×C42C22⋊D6711yescharacteristic
209C2×C42C22⋊D6711yescharacteristic
210C2×C42C22⋊D6711yescharacteristic
211C2×C42C22×D6711yescharacteristic
212C4×C8—1344no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217C24—21no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4C22⋊Dic6711yescharacteristic
228C22×C4C22⋊Dic6711yescharacteristic
229C22×C4C22⋊Dic6711yescharacteristic
230C22×C4C22⋊Dic6711yescharacteristic
231C22×C4C22×Dic6711yescharacteristic
232C22×C4C2×C22⋊D6711yescharacteristic
233C22×C4C2×C22⋊D6711yescharacteristic
234C2×C8—2682no—
235C2×C8—2682no—
236C2×C8—2682no—
237C2×C8—2682no—
238C2×C8—2682no—
239C2×C8—1344no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—41no—
256C42—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊Dic6711yescharacteristic
265Q8—82no—
266D4—82no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4#1452396012yes—
293C2×C4#1452397012yes—
294C2×C4#1452405012yes—
295C2×C4#1452406312yes—
296C2×C4#1452406312yes—
297C8—2684no—
298C8—5362no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22#1686336311yescharacteristic
307C22#1686337011yescharacteristic
308C22C2×#1452396011yescharacteristic
309C22#1686339211yescharacteristic
310C22#1686339211yescharacteristic
311C22C2×#1452397011yescharacteristic
312C22#1686341311yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—81no—
323C2—161no—
324C2#2255856111yescharacteristic
325C2#2255861711yescharacteristic
326C2#2255866911yescharacteristic
327C2#2255869511yescharacteristic
328C2#2255878611yescharacteristic
329C2#2255882011yescharacteristic
330C2#2255887111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2861C2
socleC23×C67C2×C22⋊C4
prime radicalC22×D4×C67C2×C4
cosocleC2×C42×C67C23
hypercenter#2861D67
hypocenterC67#7656
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C43 ◁ #2861

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22560352C2semidirect
2. #22559955C2semidirect
3. #22558035C2semidirect
171. C67#7656semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225603522
#225599552
#225580352
#225580302
C67×#28612
#765667